Question: The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=\frac{n+1}{n+10}$. Write a rule for ${{a}_{n}}$.
An individual term of a series is the difference between two successive partial sums. In particular, ${{a}_{n}}={{S}_{n}}-{{S}_{n-1}}\,$. $\begin{aligned} {{S}_{n}}-{{S}_{n-1}}&=\frac{n+1}{n+10}-\frac{n}{n+9} \\\\ &=\frac{(n+1)(n+9)-n(n+10)}{(n+10)(n+9)} \\\\ &=\frac{9}{(n+10)(n+9)} \end{aligned}$